Ram Karan

Some remarks on subgroups determined by certain ideals in integral group rings

LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andH∩K⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI3(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I2(G)I(H)+I(G)I(H)I(G)+I(H)I2(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH2(G/H′, T)≤1, are computed. the subgroup ofG determined byIn(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained

Augmentation quotients of integral group rings II

A description of the quotient group Δ2(G)Δ(K)Δ3(G)Δ(K) when the group G is semidirect product H ⋊ K of a normal subgroup H by a subgroup K is given. For the same group G, a nice description of one of the direct factors of the quotient group Δ2(G)Δ(H)Δ3(G)Δ(H) is also given.

A note on polynomial maps

Let ZG denote the integral group ring of a group G and Δ(G) its augmentation ideal. If H and K are subgroups of G, it is proved that G ∩ (1 + ZGΔ(H)Δ(K)) = γ2(HK ∩ K). (γi(M) denotes the ith term of the lower central series of the group M.) Also it is proved that if G> = HK, where H, K are normal subgroups of G with H ∩ K contained in the centre of G, then G ∩ (1 + Δ3(G) + Δ(H)Δ(G)) = γ2(H)γ3(G).

Augmentation Quotients of Free Group Rings

Let F be a free group and R be a subgroup of F. It is proved that are free-abelian. Explicit bases of first two and complete descriptions of all these groups are also given.

What is the probability an automorphism fixes a group element

ABSTRACT Extending the notion of probability to the automorphisms of a group, we find the probability of an arbitrarily chosen automorphism of a group fixing an arbitrary element of the group.

On equality of derival and inner automorphisms of some p-groups

For a group G, D(G) denotes the group of all derival automorphisms of G. For a finite nilpotent group of class 2, it is shown that . We prove that if G is a nilpotent group of class such that and , then if and only if . Finally, for an odd prime p, we classify all p-groups of order , for which .

Some intersections and identifications in integral group rings

LetZG be the integral group ring of a groupG and I(G) its augmentation ideal. For a free groupF andR a normal subgroup ofF, the intersectionIn+1 (F) ∩In+1 (R) is determined for alln≥ 1. The subgroupsF ∩ (1+ZFI (R) I (F) I (S)) ANDF ∩ (1 + I (R)I3 (F)) of F are identified whenR and S are arbitrary subgroups ofF.